Optimal decay–error estimates for the hyperbolic–parabolic singular perturbation of a degenerate nonlinear equation

Abstract

We consider a degenerate hyperbolic equation of Kirchhoff type with a small parameter ε in front of the second-order timederivative. In a recent paper, under a suitable assumption on initial data, we proved decay-error estimates for the difference between solutions of the hyperbolic problem and the corresponding solutions of the limit parabolic problem. These estimates show in the same time that the difference tends to zero both as ε → 0 + , and as t → +∞. In particular, in that case the difference decays faster than the two terms separately. In this paper we consider the complementary assumption on initial data, and we show that now the optimal decay-error estimates involve a decay rate which is slower than the decay rate of the two terms. In both cases, the improvement or deterioration of decay rates depends on the smallest frequency represented in the Fourier components of initial data.

On the Parabolic Regime of a Hyperbolic Equation with Weak Dissipation: The Coercive Case

“…We conclude by pointing out once again that this analysis applies to the nondegenerate coercive case. Things are quite different both in the nondegenerate noncoercive case (see [5,12,13,14,15]), and in the degenerate coercive case (see [7,8]).…”

“…The main research lines concern global existence for the parabolic and the hyperbolic problem (at least when ε is small enough), decay estimates on u(t), u ε (t), and u ε (t)−u(t) as t → +∞, error estimates on the difference as ε → 0 + , and decay-error estimates, namely estimates describing in the same time the behavior of the difference u ε (t) − u(t) as t → +∞ and ε → 0 + . The interested reader is referred to the survey [6], or to the more recent papers [2,7,8].…”

Asymptotic Limits for Mildly Degenerate Kirchhoff Equations

“…In other words, when ε is small enough solutions of (1.1) behave as solutions of the first order problem (1.6). The results proved in this paper are fundamental to provide optimal decay-error estimates for the singular perturbation problem, namely, on the difference u ε − u between solutions of (1.1) and solutions of (1.6) (see [9], [10]). Indeed in the past such decay-error estimates were known and optimal in the nondegenerate case (see [12]) but only partial results had been proved in the degenerate case (see the survey [8]).…”